Chain rule/multiple variables

1. Jun 3, 2014

chemphys1

1. The problem statement, all variables and given/known data

Show z(x,y) = cos(xy) is a solution of

(∂z/∂x)y + (∂z/dy)x = (x+y) ( (∂2z/∂x∂y) + xyz)

(question also attached if it makes it clearer)

3. The attempt at a solution

∂z= (∂z/∂x)ydx + (∂z/dy)xdy

∂z/∂x = -ysin(xy)
∂z/∂y = -xsin(xy)

what does it mean show it is a solution? any tips appreciated

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2. Jun 3, 2014

CAF123

It means that choice of z(x,y) satisfies the equation. I.e plug in z(x,y) into LHS and into the RHS and they should be equal.

3. Jun 4, 2014

Joffan

So for $z(x,y) = \cos(xy)$, what is $\frac{∂^2z}{∂x∂y}$?